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\textit{EC821:\ Time Series Econometrics, Spring 2003}

\bigskip \textit{Notes Section 1 }

\textbf{Data types}

In the narrowest sense, this course is concerned with time series
data---those in which the individual observations are indexed by some
notation of calendar time. What is a time series? In a very simple sense,
merely a set of data indexed by some regular increment of time, which may or
not be regular in the sense of the calendar. For instance, an interest rate
series containing observations from the last day of each month will be
unequally spaced in calendar time---and is likely that the ``last day'' will
be other than that in the presence of weekends and holidays. Also, we often
work with time series of ``business daily'' data, generated (at most) five
days per week. We assume that the $T$ observations available are a finite
segment of a doubly infinite sequence, which goes back into the infinite
past, and forward to infinity. We might consider several specific time
series: for instance, a time trend is merely $y_{t}=t;$ a constant series is 
$y_{t}=c,$ and a Gaussian white noise process is $y_{t}=\epsilon _{t}.$ 

We speak of time series data to contrast with the other major form of data
organization: the cross section, in which each observation is indexed by an
identifier such as a person's Social Security number, a company's CUSIP, a
country's ISO abbreviation, or a survey respondent's questionnaire ID. But
there are two other forms of data organization which we often encounter, and
which are very relevant for much of the research carried out with time
series data. These are pooled cross sections, and longitudinal or panel
data. Pooled cross sections, even if the cross sections are of the same size
(e.g., 400 respondents to a weekly poll of presidential popularity) do not
give rise directly to time series, since the individual observations are not
linked across time. However, summary statistics from each of these cross
sections may be computed, and assembled into time series to illustrate
temporal changes in the sample's measures of central tendency and dispersion.

One of the most rapidly expanding areas in applied econometrics is the use
of longitudinal, or panel, data---data indexed by both $i$ and $t$
subscripts, essentially repeated measurements on the same individuals over
time. Panel data in common use in economics and finance have tended to
appear in two common forms: the ``small T, large N'' panel, exemplified by
Compustat firm--level data, or the Panel Study of Income Dynamics household
data. These datasets have relatively few time--series observations but
thousands of individuals. Many of the econometric techniques developed in
this area make use of the ``small T, large N'' setting: for instance,
Arellano and Bond's dynamic panel data GMM\ estimation technique (cf. Stata
command \texttt{xtabond}). The econometric theory underlying these
estimators is based on N$\rightarrow \infty $ for T fixed. In this context,
there are few time--series aspects of the data that may be modelled, since
the number of time series observations is quite limited. The other common
form is the ``small N, large T'' panel, exempified by daily data from the
financial markets (G--7 exchange rates or inflation rates, stock price
series for a limited number of firms, etc.) In this context, we generally
have hundreds or thousands of time series observations on each unit, but a
relatively small number of units. This allows many more of the time series
properties of the data to be exploited in the estimation. Similar to the
first form, the underlying asymptotic theory for estimators applied to these
data is based on the assumption that T$\rightarrow \infty $ for N fixed.
Datasets of these form are often used to perform panel unit root tests, or
analyses of cointegration. 

\textbf{Dimensionality}

When we utilise time series data, it is important to determine the
dimensionality of a variable: e.g., whether it refers to a level, stock,
flow, or rate of change. An interest rate, or a consumer price index, or an
index of industrial production at a point in time is a level. The capital
stock employed by the firms in an industry at a point in time is just
that---a stock, in units of number of machines, or their value in real
terms. That stock is accumulated or decumulated over time via capital
investment, which is a flow---and must refer to the unit of time over which
the flow is measured. If a flow is converted into percentage terms, it
becomes a dimensionless rate of change, such as the rate of change of the
consumer price index, which we term inflation. It is customary to measure
rates of change at an annual rate, so that inflation might be described as
reaching three percent last month: for most economies, this does not reflect
the monthly change in the price level, but rather the monthly change
compounded to an annual rate. 

\textbf{Time series operators}

We make extensive use in time series models of various time series
operators, as shorthand to specify how a variable may be transformed. For
instance, the multiplication operator might be used: $y_{t}=\beta x_{t}.$
Unlike ordinary algebraic manipulation, this represents applying the
operator to each element of the time series (for all defined $t),$ analogous
to filling a formula through all the rows of a spreadsheet, or using Stata's 
\texttt{generate} command to define a mappng between $x$ and $y.$ Since the
multiplication or addition operators refer to element--wise operations, they
follow all the rules of algebra: e.g. $z_{t}=\beta \left( y_{t}+x_{t}\right) 
$ can be implemented either by adding each period's values and multiplying
the result by $\beta ,$ or by multiplying through the parens and then
summing. Multiplication and addition are commutative. 

The commonly employed first difference operator, $\Delta ,$ transforms a
series into its increments: for instance, $\Delta K_{t}$, the first
difference of the capital stock, will by convention generate the change
between the previous period's value, $K_{t-1},$ and the current value, $%
K_{t}.$ This involves computing the lagged value of $K_{t},$ which may
itself be expressed using the lag operator as $LK_{t}:$ that is, the lag
operator applied to a set of observations ``backshifts'' them by one time
unit. For this reason, many textbooks and articles express $L$ as the $B,$
or backshift operator. Note that the lag operator may be expressed as $L^{j};
$ if $j=1,$ we consider the previous period's lag, but may express any lag.
For quarterly data, $L^{4}y_{t}$ would express the value of $y$ four
quarters ago, while the lead of $y,$ $L^{-4}y_{t},$ would refer to the value
of $y$ four periods later. A negative power applied to the lag generates the
lead, and just as any algebraic quantity raised to the zero power is itself, 
$L^{0}y_{t}=y_{t}.$ The lag operator also satisfies the law of exponents, so
that $L^{i}L^{j}y_{t}=L^{i+j}y_{t}=y_{t-(i+j)};$ it is commutative with the
multiplication operator, and distributive over the addition operator, so
that $L\left( y_{t}+x_{t}\right) =Ly_{t}+Lx_{t}.$ 

One must be cautious in using the difference operator to note that $\Delta %
^{j}$ refers to the $j^{th}$difference:\ that is, the difference operator
applied repeatedly. Thus $\Delta y_{t}=y_{t}-y_{t-1,}$ and $\Delta ^{2}y_{t}=%
\Delta \Delta y_{t}=$ $\left( y_{t}-y_{t-1}\right) -\left(
y_{t-1}-y_{t-2}\right) =y_{t}-2y_{t-1}+y_{t-2},$ and so on. One may define $%
\Delta ^{j}$ for any positive $j,$ but this should not be mistaken for,
e.g., $y_{t}-L^{4}y_{t},$ which is not the ``fourth difference'' of $y_{t}.$
It should be remembered that the difference operator is the discrete
equivalent of $\frac{dy}{dt},$ so that the second difference is the analog
to $\frac{d^{2}y}{dt^{2}}:$ the acceleration of $y.$ 

In Stata, the time series operators $D.,$ $L.$ and $F.$ when prefixing a
variable are the first difference, lag, and lead (forward) of the variable,
respectively. The number of times that the operator is to be applied may
also be specified in this syntax, so that one may say $L4.y$ or $D2.price$
to refer to the fourth lag of $y,$ or the second difference of the price
level, respectively.

\textbf{Lag polynomials}

In working with these time series operators, we often employ lag
polynomials; e.g. the difference operator $\Delta $ may be defined as $%
\left( 1-L\right) ,$ a first--order polynomial in the lag operator, while $%
\Delta ^{2}=(1-L)^{2}.$ This allows easier access to the definition of the
higher--order difference; since the expression  $(1-L)^{2}$ may be expanded
as $(1-2L+L^{2}),$ it may immediately be seen that  $\Delta %
^{2}y_{t}=y_{t}-2y_{t-1}+y_{t-2},$ where the coefficients of the expanded
lag polynomial appear in each term. As shorthand, we often will write $%
A(L)=(1-\phi _{1}L-\phi _{2}L^{2}-\phi _{3}L^{3})$ as a finite--order lag
polynomial The shorthand $A(1)$ refers to the steady state representation,
in which all lags are set to zero:\ $A(1)=(1-\sum \phi _{i}).$ We might also
write a factored polynomial, such as $\left( 1-\lambda _{1}L\right) $ $%
\left( 1-\lambda _{2}L\right) x_{t}=x_{t}-\left( \lambda _{1}+\lambda
_{2}\right) x_{t-1}+\lambda _{1}\lambda _{2}x_{t-2}.$

\textbf{\ The univariate dynamic model}

The simplest model of a stochastic process $y_{t}$ could be written as a
linear univariate dynamic model 

\[
y_{t}=\phi _{1}y_{t-1}+\phi _{2}y_{t-2}+...+\phi _{p}y_{t-p}+\epsilon _{t}
\]
which is a $p^{th}$ order stochastic difference equation with constant
coefficients. The process is said to be autoregressive, since the value of $y
$ depends on its own past values, as well as the innovation process $%
\epsilon .$ This may be written in lag polynomial notation as $\phi \left(
L\right) y_{t}=\epsilon _{t}$ where the polynomial will be $\left( 1-\phi
_{1}L+\phi _{2}L^{2}+...+\phi _{p}L^{p}\right) .$ We will consider this
model (or its more general counterpart which incorporates a constant term)
as an $AR(p)$ model of $y_{t}.$

\textbf{A dynamic bivariate model}

Let us consider a dynamic model that contains two variables: an
implementation of the partial adjustment model (PAM) that might be applied
to the stock of consumer durables, or a firm's capital stock. The PAM
contains a target value for $y_{t},$ denoted $y_{t}^{*},$ which in
equilibrium will be realised. Due to adjustment costs, economic agents do
not reach equilibrium immediately following a revision of the target value.
Thus the law of motion for $y_{t}$ may be expressed as
\[
\Delta y_{t}=y_{t}-y_{t-1}=\delta \left( y_{t}^{*}-y_{t-1}^{{}}\right)
+\epsilon _{t},\left| \delta \right| <1
\]
where the parameter $\delta $ indicates the speed of adjustment; the closer
is $\delta $ to unity, the more rapidly is equilibrium restored. The model
may be rewritten as
\begin{equation}
y_{t}=\delta y_{t}^{*}+\left( 1-\delta \right) y_{t-1}^{{}}+\epsilon _{t}
\label{pam2}
\end{equation}
so that the current value of $y_{t}$ is a convex combination of the desired
stock and last period's value. If we now define the desired stock as
depending upon an exogenous factor $x_{t},$ $y_{t}^{*}=\lambda _{0}x_{t},$
that may be substituted into (\ref{pam2}) to yield, with some redefinition
of parameters,
\[
\left( 1-\gamma _{1}L\right) y_{t}=\beta _{0}x_{t}+\epsilon _{t}
\]
The resulting dynamic model has one lag on the dependent variable and no
lags on the explanatory variable. The model may be generalised, of course,
to include richer dynamics in the determination of the desired stock.

\textbf{The autoregressive distributed lag (ADL)\ model\footnote{%
This section is taken from Davidson and MacKinnon, Estimation and Inference
in Econometrics, section 19.4.} }

The model as given above is a special case of the autoregressive distributed
lag (ADL)\ model. We can express the general ADL\ model as $ADL(p,q):$%
\begin{eqnarray*}
\left( 1-\sum_{i=1}^{p}\gamma _{i}L^{i}\right) y_{t} &=&\alpha _{0}+\left(
1-\sum_{j=0}^{q}\beta _{j}L^{j}\right) x_{t}+\epsilon _{t} \\
\gamma \left( L\right) y_{t} &=&\alpha _{0}+\beta \left( L\right)
x_{t}+\epsilon _{t}
\end{eqnarray*}
where $\epsilon _{t}$ is assumed to be IID(0,$\sigma ^{2}).$ If we consider
the $ADL(1,1)$ special case,

\begin{equation}
y_{t}=\alpha _{0}+\gamma _{1}y_{t-1}+\beta _{0}x_{t}+\beta
_{1}x_{t-1}+\epsilon _{t}  \label{adl11}
\end{equation}
we may note that many models are nested within it. For instance, if $\gamma
_{1}=\beta _{1}=0,$ we have a static regression. A univariate $AR(1)$ model
for $y$ sets $\beta _{0}=\beta _{1}=0.$ A model in first differences is a
special case with $\gamma _{1}=1$ and $\beta _{1}=-\beta _{0}$; $\alpha _{0}$
in that context implies the absence of a time trend. Since all of these
models are nested within the $ADL(1,1)$ model, they may readily be tested as
restrictions upon that more general form. 

We may also consider the steady state of this model, and the long run effect
of $x$ upon $y.$ By removing all time subscripts and solving, we may derive
that 
\begin{equation}
y=\frac{\alpha _{0}}{1-\gamma _{1}}+\lambda x,~\lambda =\frac{\beta
_{0}+\beta _{1}}{1-\gamma _{1}}  \label{adlss}
\end{equation}
where the long run multiplier contains the sum of the lag coefficients,
amplified by the term $\left( 1-\gamma _{1}\right) ^{-1}.$ The stability
condition requires that $\left| \gamma _{1}\right| <1.$ If $y$ and $x$ were
measured in logarithms (as often is the case in macroeconomic models) then $%
\lambda $ will be a long--run elasticity.

An interesting feature of the ADL\ model is that it may be rewritten in many
different forms without affecting the model's ability to explain the data,
or changing the least squares estimates of the coefficients of interest. For
instance, we may rewrite (\ref{adl11})\ as:
\begin{eqnarray*}
\Delta y_{t} &=&\alpha _{0}+\left( \gamma _{1}-1\right) y_{t-1}+\beta
_{0}x_{t}+\beta _{1}x_{t-1}+\epsilon _{t} \\
\Delta y_{t} &=&\alpha _{0}+\left( \gamma _{1}-1\right) y_{t-1}+\beta _{0}%
\Delta x_{t}+\left( \beta _{0}+\beta _{1}\right) x_{t-1}+\epsilon _{t} \\
\Delta y_{t} &=&\alpha _{0}+\left( \gamma _{1}-1\right) y_{t-1}-\beta _{1}%
\Delta x_{t}+\left( \beta _{0}+\beta _{1}\right) x_{t}+\epsilon _{t} \\
\Delta y_{t} &=&\alpha _{0}+\left( \gamma _{1}-1\right) \left(
y_{t-1}-\lambda x_{t-1}\right) +\beta _{0}\Delta x_{t}+\epsilon _{t}
\end{eqnarray*}

These reformulations are useful, in that they make it possible to receive
direct point and interval estimates of, say, the sum of the coefficients on
x (as in the second and third forms). The most celebrated form is the last,
which is known as the error correction form or error correction model (ECM),
as defined by Hendry and Anderson (1977) and Davidson et al. (DHSY, 1978).
The ECM, as we shall later discuss, expresses the revision of $y$ in terms
of the most recently observed disequilibrium in the system---since the
parenthesized expression is the degree to which the long run equilibrium of
the system was perturbed in the previous time period. Unlike the other forms
of the model, the ECM\ form introduces a nonlinearity into the model, but it
merely represents a linear model reparameterized in a nonlinear fashion. The
error correction term is implicitly present in any of the other
specifications, since its coefficient may be recovered from all of them. If
the restriction $\lambda =1$ is imposed (which may be sensible if $y$ and $x$
are similar in magnitude), the ECM form becomes linear in the parameters.
This restriction, given the long run multiplier, is equivalent to the
restriction that  $\gamma _{1}+\beta _{0}+\beta _{1}=1,$ which may readily
be tested from any of the other forms of the model.

An excellent reference for these topics is Hendry, Pagan, Sargan, \textit{%
Dynamic specification}, chapter 18 in the \textit{Handbook of Econometrics},
volume II (1984). 

\textbf{Vector time series models}

These ADL\ models may be straightforwardly extended to the multivariate
context by considering $y_{t}$ as a vector of several variables, which then
are determined by their own lags as well as potentially influenced by the
current and lagged values of additional variables. A pure autoregressive
form of this structure is the $VAR$, or vector autoregression, which we will
discuss at length later in the course. The VAR\ model may be augmented with
additional exogenous variables, in which case the counterpart to the steady
state of the univariate ADL\ model (\ref{adlss})\ may be defined in terms of
a matrix lag polynomial. In the simplest case---that of a two--variable
``pure VAR''---we have $y_{t}=(y_{1t},y_{2t})^{^{\prime }}.$ If we consider
a fourth--order VAR, we may write 
\[
\left( I-\Pi _{1}L-\Pi _{2}L^{2}-\Pi _{3}L^{3}-\Pi _{4}L^{4}\right)
y_{t}=\mu +\epsilon _{t}
\]
where $\mu $ is a 2x1 vector of constant terms, and $\epsilon $ is a 2x1
vector of error terms. This model can be rewritten as

\[
A(L)y_{t}=\mu +\epsilon _{t}
\]
Just as in the univariate case, we may evaluate the polynomial at $A(1)$ in
order to compute the sum of the lag weights. 

We may also consider a VAR\ with additional exogenous variables---the
multivariate analogue to the ADL\ model---in which an additional polynomial $%
B(L)x_{t}$ appears, so that now
\[
A(L)y_{t}=\mu +B(L)x_{t}+\epsilon _{t}
\]

The long-run value of the system may be computed, assuming that the $A(L)\;$%
polynomial is invertible, as

\[
y_{t}=A(L)^{-1}\mu +A(L)^{-1}B(L)x_{t}+A(L)^{-1}\epsilon _{t}
\]

which is sometimes termed the final form of the system. This in turn may be
expressed as

\[
y_{t}=\mu ^{*}+C(L)x_{t}+\upsilon _{t}
\]

where the polynomial $C(L)\;$is a ``rational lag''---a ratio of two
finite-order polynomials, which is an expression of infinite order, $%
C(L)=\sum_{i=0}^{\infty }C_{i}L^{i}.$ The long-run relationship is given by $%
C(1)=A(1)^{-1}B(1),$ which will only be defined when the matrix $A(1)$ is
non-singular. This long--run equilibrium may be achieved iff the model is
dynamically stable; for a vector process, the requirement is that the
eigenvalues of the matrix polynomial $A(L)$ have modulus less than one in
absolute value. Since $A(L)$ is not in general a symmetric matrix, its
eigenvalues will be complex, so that the moduli of these complex quantities
must be evaluated (see Stata routine \texttt{geneigen}).   

We will next take up the definitions of stationary and nonstationary random
variables. 

\end{document}
